Finite Group Databasebeta

← #13259271

Subgroups — #13259271

Order 1792.

1230 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6470C211yesmaximal, characteristic
2C2×#89905C211yesmaximal, characteristic
3C2×#89738C211yesmaximal, characteristic
4C2×C7×#343C211yesmaximal, characteristic, complemented
5C2×#89353C211yesmaximal, characteristic, complemented
6C2×#89349C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8C22×#1157C2211yescharacteristic
9C22×C7×M16C2211yescharacteristic
10C22×C7×M16C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic7C2211yescharacteristic
13C22×#1136C2211yescharacteristic
14C23×C7⋊C8C2211yescharacteristic
15C2×#6470—22no—
16C2×#6470—22no—
17C2×#6470—22no—
18C2×#6470—22no—
19C2×#6470—22no—
20C2×#6470C414yescomplemented
21C22×#1131—21no—
22C22×C8×D7—21no—
23#89905—22no—
24#89905C414yescomplemented
25C2×#6395—21no—
26C2×#6390—21no—
27C2×#6384—21no—
28#89738—22no—
29#89738C414yescomplemented
30C2×#6383—21no—
31C7×#343—22no—
32#89353—22no—
33#89349—22no—
34#89324—21no—
35#89324—21no—
36#10774—71nomaximal, Sylow
37C23×C4×C7—21no—
38C23×C4×C7C2311yescharacteristic
39C22×C22⋊D7—21no—
40C23×Dic7—21no—
41C2×#1157—21no—
42C2×#1157—21no—
43C2×#1157—21no—
44C2×#1157—21no—
45C2×#1157—21no—
46C2×#1157—21no—
47C2×#1157C2×C412yes—
48C2×#1157C2×C412yes—
49C2×#1157C2×C412yes—
50C2×#1157D412yes—
51C2×#1157D412yes—
52C2×#1157Q812yes—
53C22×D28D411yescharacteristic
54C22×C4×D7—21no—
55C22×Dic14D411yescharacteristic
56C2×C7×M16—21no—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C2×C7×M16—21no—
60C2×C7×M16—21no—
61C2×C7×M16—21no—
62C2×C7×M16—21no—
63C2×C7×M16—21no—
64C2×C7×M16C2×C412yes—
65C2×C7×M16C2×C412yes—
66C2×C7×M16D412yes—
67C2×C7×M16D412yes—
68C22×C7×C8—21no—
69C22×C7×C8—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2×C412yes—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2×C412yes—
82C2×#1136—21no—
83C2×#1136—21no—
84C2×#1136—21no—
85C2×#1136—21no—
86C2×#1136—21no—
87C2×#1136C2×C412yes—
88C22×C7⋊C8—21no—
89C22×C7⋊C8—21no—
90C22×C7⋊C8—21no—
91C22×C7⋊C8—21no—
92C22×C7⋊C8—21no—
93C22×C7⋊C8C2×C412yes—
94C22×C7⋊C8D412yes—
95C22×C7⋊C8D411yescharacteristic
96C22×C7⋊C8D411yescharacteristic
97#6470—28no—
98#6470—28no—
99#6470—28no—
100#6470—28no—
101C2×#1131—24no—
102C2×#1131—24no—
103C2×#1131—24no—
104C2×C8×D7—24no—
105C2×C8×D7—24no—
106C2×C8×D7—24no—
107#6395—24no—
108#6390—24no—
109#6384—24no—
110#6383—24no—
111C22×#131—71no—
112C2×#417—71no—
113C2×#417—71no—
114C2×#343—71no—
115C2×#343D711yescharacteristic, complemented
116C2×#341—71no—
117C2×#340—71no—
118C24×C7#4111yescharacteristic
119C23×D7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—41no—
122C22×C4×C7—41no—
123C22×C4×C7—81no—
124C22×C4×C7—81no—
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×Q811yescharacteristic
136C22×C4×C7#4111yescharacteristic
137C22×C4×C7#4111yescharacteristic
138C22×C4×C7#4111yescharacteristic
139C2×C22⋊D7—22no—
140C2×C22⋊D7—22no—
141C2×C22⋊D7—22no—
142C2×C22⋊D7—22no—
143C2×C22⋊D7—22no—
144C2×C22⋊D7—22no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—41no—
148C22×Dic7—41no—
149C22×Dic7—81no—
150C22×Dic7—81no—
151#1157—22no—
152#1157—22no—
153#1157—22no—
154#1157—22no—
155#1157—22no—
156#1157—22no—
157#1157—22no—
158#1157—22no—
159#1157—22no—
160#1157—22no—
161#1157—22no—
162#1157—22no—
163#1157C4214yes—
164#1157C22⋊C414yes—
165#1157C4⋊C414yes—
166#1157C4⋊C414yes—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28C22⋊C412yes—
171C2×D28C22⋊C412yes—
172C2×D28C4⋊C412yes—
173C2×C4×D7—22no—
174C2×C4×D7—22no—
175C2×C4×D7—22no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—22no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14C22⋊C412yes—
183C2×Dic14C22⋊C412yes—
184C2×Dic14C4⋊C412yes—
185C7×M16—22no—
186C7×M16—22no—
187C7×M16—22no—
188C7×M16—22no—
189C7×M16—22no—
190C7×M16—41no—
191C7×M16—41no—
192C7×M16C22⋊C414yes—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—41no—
198C2×C7×C8—41no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—41no—
202C7×C4⋊C4—41no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4⋊Dic7—41no—
206C4⋊Dic7—41no—
207#1136—22no—
208#1136—22no—
209#1136—41no—
210#1136—41no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—22no—
218C2×C7⋊C8—22no—
219C2×C7⋊C8—22no—
220C2×C7⋊C8—41no—
221C2×C7⋊C8—41no—
222C2×C7⋊C8C22⋊C412yes—
223C2×C7⋊C8C22⋊C412yes—
224#1131—216no—
225C8×D7—216no—
226C22×#41—71no—
227C2×#131—74no—
228C2×#131—142no—
229C2×#131—142no—
230C2×#131—142no—
231C2×#131—142no—
232C2×#131—142no—
233C22×M16—71no—
234C22×M16—141no—
235C22×M16C2×D711yescharacteristic
236C22×M16C2×D711yescharacteristic
237C23×C8—71no—
238C23×C8—141no—
239C22×C4⋊C4—71no—
240C22×C4⋊C4C2×D711yescharacteristic
241#417—74no—
242#417—74no—
243#417—142no—
244#417—142no—
245C2×#103—141no—
246C2×#103—141no—
247C2×#102—141no—
248C2×#102—141no—
249#343—22no—
250#343—142no—
251#341—142no—
252#340—141no—
253#340—141no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7#11711yescharacteristic
259C23×C7C4×D411yescharacteristic
260C23×C7C4×D411yescharacteristic
261C23×C7#12111yescharacteristic
262C23×C7C22⋊Q811yescharacteristic
263C23×C7#12311yescharacteristic
264C23×C7#12311yescharacteristic
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—82no—
274C2×C4×C7—81no—
275C2×C4×C7—81no—
276C2×C4×C7—82no—
277C2×C4×C7C2×C4211yescharacteristic
278C2×C4×C7C2×C22⋊C411yescharacteristic
279C2×C4×C7C2×C22⋊C411yescharacteristic
280C2×C4×C7C2×C22⋊C411yescharacteristic
281C2×C4×C7C2×C22⋊C411yescharacteristic
282C2×C4×C7C2×C22⋊C411yescharacteristic
283C2×C4×C7C2×C4⋊C411yescharacteristic
284C2×C4×C7C2×C4⋊C411yescharacteristic
285C2×C4×C7C2×C4⋊C411yescharacteristic
286C2×C4×C7#11711yescharacteristic
287C2×C4×C7C4×D411yescharacteristic
288C2×C4×C7C4×D411yescharacteristic
289C2×C4×C7C4×D411yescharacteristic
290C2×C4×C7C4×D411yescharacteristic
291C2×C4×C7C4×D411yescharacteristic
292C2×C4×C7C4×D411yescharacteristic
293C2×C4×C7#12011yescharacteristic
294C2×C4×C7#12011yescharacteristic
295C2×C4×C7#12011yescharacteristic
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12111yescharacteristic
298C2×C4×C7#12111yescharacteristic
299C2×C4×C7#12111yescharacteristic
300C2×C4×C7C22⋊Q811yescharacteristic
301C2×C4×C7C22⋊Q811yescharacteristic
302C2×C4×C7C22⋊Q811yescharacteristic
303C2×C4×C7#12311yescharacteristic
304C2×C4×C7#12311yescharacteristic
305C22⋊D7—24no—
306C22⋊D7—24no—
307C22⋊D7—24no—
308C22⋊D7—24no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—42no—
313C2×Dic7—82no—
314C2×Dic7—81no—
315C2×Dic7—81no—
316C2×Dic7—82no—
317D28—22no—
318D28—22no—
319D28—22no—
320D28#9514yes—
321C4×D7—24no—
322C4×D7—24no—
323C4×D7—24no—
324C4×D7—24no—
325Dic14—22no—
326Dic14—22no—
327Dic14—22no—
328Dic14#9514yes—
329C7×C8—24no—
330C7×C8—42no—
331C7⋊C8—22no—
332C7⋊C8—22no—
333C7⋊C8—42no—
334C2×#41—72no—
335C2×#41—72no—
336C2×#41—72no—
337C2×#41—72no—
338C2×#41—72no—
339C2×#41—72no—
340C2×#41—141no—
341C2×#41—141no—
342C2×#41—141no—
343C2×#41—141no—
344C2×#41—141no—
345C2×#41—141no—
346C22×Q8—71no—
347C22×D4—71no—
348C22×D4—141no—
349C23×C4—21no—
350C23×C4—141no—
351C23×C4—141no—
352C23×C4C22×D711yescharacteristic
353#131—148no—
354#131—148no—
355#131—148no—
356#131—148no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—72no—
366C2×M16—144no—
367C2×M16—144no—
368C2×M16—144no—
369C2×M16—141no—
370C2×M16—141no—
371C2×M16—141no—
372C2×M16—141no—
373C2×M16—141no—
374C2×M16C4×D712yes—
375C2×M16C4×D712yes—
376C2×M16D2812yes—
377C2×M16C22⋊D712yes—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—72no—
381C22×C8—71no—
382C22×C8—71no—
383C22×C8—72no—
384C22×C8—144no—
385C22×C8—144no—
386C22×C8—144no—
387C22×C8—141no—
388C22×C8—141no—
389C22×C8—141no—
390C22×C8—141no—
391C22×C8—141no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—72no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4C4×D712yes—
404#103—144no—
405#103—144no—
406#102—144no—
407#102—144no—
408C22×C7—21no—
409C22×C7—21no—
410C22×C7—21no—
411C22×C7—21no—
412C22×C7—21no—
413C22×C7—21no—
414C22×C7—21no—
415C22×C7—21no—
416C22×C7—21no—
417C22×C7—21no—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7C4×C22⋊C411yescharacteristic
421C22×C7#38011yescharacteristic
422C22×C7#38111yescharacteristic
423C22×C7#38511yescharacteristic
424C22×C7#38511yescharacteristic
425C22×C7#38611yescharacteristic
426C22×C7#38611yescharacteristic
427C22×C7#47111yescharacteristic
428C22×C7#47111yescharacteristic
429C22×C7#47211yescharacteristic
430C22×C7#47311yescharacteristic
431C2×D7—24no—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—42no—
436C2×D7—42no—
437C2×D7—42no—
438C4×C7—82no—
439C4×C7C2×#9511yescharacteristic
440C4×C7C4×C22⋊C411yescharacteristic
441C4×C7#37911yescharacteristic
442C4×C7#38011yescharacteristic
443C4×C7#38511yescharacteristic
444C4×C7#38511yescharacteristic
445C4×C7#38611yescharacteristic
446C4×C7#38611yescharacteristic
447Dic7—24no—
448Dic7—82no—
449C24—141no—
450C24#115711yescharacteristic
451#41—74no—
452#41—74no—
453#41—74no—
454#41—74no—
455#41—142no—
456#41—142no—
457#41—142no—
458#41—142no—
459#41—142no—
460#41—142no—
461#41—142no—
462#41—142no—
463#41—142no—
464#41—142no—
465#41—142no—
466#41—142no—
467C2×Q8—72no—
468C2×Q8—72no—
469C2×Q8—72no—
470C2×Q8—141no—
471C2×Q8—141no—
472C2×Q8—141no—
473C2×D4—72no—
474C2×D4—72no—
475C2×D4—72no—
476C2×D4—142no—
477C2×D4—142no—
478C2×D4—142no—
479C2×D4—142no—
480C2×D4—142no—
481C2×D4—142no—
482C2×D4—141no—
483C2×D4—141no—
484C2×D4—141no—
485C22×C4—22no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—141no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—281no—
499C22×C4—281no—
500C22×C4—561no—
501C22×C4—561no—
502C22×C4C2×C4×D711yescharacteristic
503C22×C4C2×C4×D711yescharacteristic
504C22×C4C2×C4×D711yescharacteristic
505C22×C4C2×D2811yescharacteristic
506C22×C4#115711yescharacteristic
507C22×C4D4×D711yescharacteristic
508C22×C4D4×D711yescharacteristic
509C22×C4D4×D711yescharacteristic
510C22×C4D4×D711yescharacteristic
511C22×C4D4×D711yescharacteristic
512C22×C4#115911yescharacteristic
513C22×C4Q8×D711yescharacteristic
514C22×C4#116111yescharacteristic
515C22×C4C2×C22⋊D711yescharacteristic
516M16—22no—
517M16—22no—
518M16—22no—
519M16—22no—
520M16—22no—
521M16—41no—
522M16—41no—
523M16—1416no—
524M16—142no—
525M16—142no—
526M16—281no—
527M16—281no—
528M16(C2×D7)⋊C414yes—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—41no—
534C2×C8—41no—
535C2×C8—72no—
536C2×C8—72no—
537C2×C8—1416no—
538C2×C8—142no—
539C2×C8—141no—
540C2×C8—141no—
541C2×C8—142no—
542C2×C8—142no—
543C2×C8—141no—
544C2×C8—141no—
545C2×C8—142no—
546C2×C8—142no—
547C2×C8—281no—
548C2×C8—281no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—142no—
554C4⋊C4—142no—
555C4⋊C4—281no—
556C4⋊C4—281no—
557C2×C7—21no—
558C2×C7—21no—
559C2×C7—21no—
560C2×C7—21no—
561C2×C7C22⋊#9511yescharacteristic
562C2×C7#181511yescharacteristic
563C2×C7#181611yescharacteristic
564C2×C7#190711yescharacteristic
565C2×C7#190711yescharacteristic
566C2×C7#190811yescharacteristic
567C2×C7#190811yescharacteristic
568D7—44no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—142no—
574C23—142no—
575C23—142no—
576C23—281no—
577C23—281no—
578C23—281no—
579C23—281no—
580C23#643511yescharacteristic
581C23C4×D2811yescharacteristic
582C23#644611yescharacteristic
583C23#644711yescharacteristic
584C23#645711yescharacteristic
585C23#645911yescharacteristic
586C23C4×C22⋊D711yescharacteristic
587Q8—74no—
588Q8—142no—
589Q8—142no—
590Q8—142no—
591D4—74no—
592D4—144no—
593D4—144no—
594D4—144no—
595D4—144no—
596D4—142no—
597D4—142no—
598D4—142no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—142no—
605C2×C4—142no—
606C2×C4—142no—
607C2×C4—144no—
608C2×C4—144no—
609C2×C4—144no—
610C2×C4—144no—
611C2×C4—282no—
612C2×C4—562no—
613C2×C4—561no—
614C2×C4—561no—
615C2×C4—562no—
616C2×C4C42×D711yescharacteristic
617C2×C4C4×D2811yescharacteristic
618C2×C4D7×C22⋊C411yescharacteristic
619C2×C4D7×C22⋊C411yescharacteristic
620C2×C4D7×C22⋊C411yescharacteristic
621C2×C4D7×C22⋊C411yescharacteristic
622C2×C4(C22⋊D7)⋊C411yescharacteristic
623C2×C4(C22⋊D7)⋊C411yescharacteristic
624C2×C4#644511yescharacteristic
625C2×C4#644511yescharacteristic
626C2×C4#644611yescharacteristic
627C2×C4#644711yescharacteristic
628C2×C4D7×C4⋊C411yescharacteristic
629C2×C4D7×C4⋊C411yescharacteristic
630C2×C4D7×C4⋊C411yescharacteristic
631C2×C4#645511yescharacteristic
632C2×C4#645611yescharacteristic
633C2×C4#645611yescharacteristic
634C2×C4#645711yescharacteristic
635C2×C4#645811yescharacteristic
636C2×C4#645911yescharacteristic
637C2×C4#646011yescharacteristic
638C2×C4C2×(C2×D7)⋊C411yescharacteristic
639C2×C4C4×C22⋊D711yescharacteristic
640C2×C4#650011yescharacteristic
641C2×C4#650011yescharacteristic
642C2×C4#650111yescharacteristic
643C2×C4#650911yescharacteristic
644C8—24no—
645C8—42no—
646C8—142no—
647C8—142no—
648C8—282no—
649C7#1077411yescharacteristic, complemented, Sylow
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—144no—
663C22—282no—
664C22—282no—
665C22—282no—
666C22—282no—
667C22—282no—
668C22—282no—
669C22#8943611yescharacteristic
670C22#8943711yescharacteristic
671C22#8943911yescharacteristic
672C22#8944011yescharacteristic
673C22#8952011yescharacteristic
674C22#8952111yescharacteristic
675C22#8952611yescharacteristic
676C22#8952711yescharacteristic
677C22C4×(C2×D7)⋊C411yescharacteristic
678C22#8972811yescharacteristic
679C22#8976111yescharacteristic
680C4—82no—
681C4—144no—
682C4—562no—
683C4D7×#9511yescharacteristic
684C4#8943711yescharacteristic
685C4#8943811yescharacteristic
686C4#8944011yescharacteristic
687C4#8972711yescharacteristic
688C4#8972811yescharacteristic
689C4#8975911yescharacteristic
690C4#8976111yescharacteristic
691C2—21no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—284no—
696C2#1172675411yescharacteristic
697C2#1172675511yescharacteristic
698C2#1172676011yescharacteristic
699C2#1172676111yescharacteristic
700C2(C2×D7)⋊#9511yescharacteristic
701C2#1172784211yescharacteristic
702C2#1172787711yescharacteristic
subgroupG/H
centerC23#6447
commutatorC2×C4×C7C2×C42
FrattiniC23×C4C22×D7
FittingC2×C7×#343C2
socleC23×C7#121
prime radicalC22×#1157C22
cosocleC23×C4×C7C23
hypercenterC2×#343D7
hypocenterC7#10774
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×C7×#343C2semidirect
5. C2×#89353C2semidirect
6. C2×#89349C2semidirect
7. C2×#89324C2semidirect
20. C2×#6470C4semidirect
24. #89905C4semidirect
29. #89738C4semidirect
115. C2×#343D7semidirect
649. C7#10774semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#64702
C2×#899052
C2×#897382
C2×C7×#3432
C2×#893532
C2×#893492
C2×#893242
#107747